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Find minimum total range-increment to sort

AmazonSoftware Engineer · Online Assessment
MediumSoftware EngineerOnline AssessmentCoding & Algorithms

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This problem tests your ability to determine the minimum total operations required to make an array non-decreasing by incrementing contiguous subarrays, requiring insights into difference arrays or monotonic properties.

Problem Statement

You are given an array arr of n integers. You can perform an operation: choose a range [i, j] (inclusive) and increment all elements in that range by 1. Each operation costs 1 unit. Your goal is to find the minimum number of operations required to make the array sorted in non-decreasing order (i.e., arr[i] int:

Constraints

  • 1 <= n <= 10^5
  • 0 <= arr[i] <= 10^9

Examples

Example 1: Input: arr = [3, 2, 1] Output: 2 Explanation:

  1. Increment [1, 2] to get [3, 3, 2].
  2. Increment [2, 2] to get [3, 3, 3]. Alternatively, identify the required differences: d[i] = arr[i] - arr[i-1]. If d[i] < 0, we need to cover the deficiency.

Example 2: Input: arr = [1, 2, 3] Output: 0 Explanation: The array is already sorted.

Edge Cases

  • Array with 1 element: Output should be 0.
  • Array with all identical elements: Output should be 0.
  • Strictly decreasing array: The algorithm must handle the cumulative needs.

Hints

Hint 1

Consider the difference array `diff[i] = arr[i] - arr[i-1]` for `i > 0`.

Hint 2

The condition `arr[i] >= arr[i-1]` is equivalent to `arr[i] - arr[i-1] >= 0`.

Hint 3

If `arr[i] < arr[i-1]`, you have a 'deficit' that must be corrected by increments applied to the suffix starting at index `i`.

Follow-up questions

  • How does the complexity change if you are allowed to increment or decrement the range?
  • What if the cost of an operation is not 1, but proportional to the length of the range `(j - i + 1)`?
  • Can you solve this problem if you are only allowed to perform operations on a fixed-length window of size `k`?